Turtle of Lo Shu embodying traditional magic squares

Constrained, Unconstrained and Surprised in a Global Context (Part #10)

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]

As explored by Tao Jiang in his comments on the Zhuangzi, reference could have been made to the acclaimed importance of the turtle in the period when the Zhuangzi is believed to have been composed. This derives from the tradition of the magical Lo Shu turtle which had emerged from a river at the time of a great flood, as observed by the legendary Emperor Yu. The strange markings on its back were allegedly associated by the Emperor Fuxi (or Fu Hsi) with the form of a magic square (below centre). Hence the considerable subsequent commentary and depiction of such a turtle in relation to the Lo Shu magic square and as fundamental to feng shui (summarized by the correspondences, below right).

Magic squares: The correspondences in the image on the right below feature in an extensive discussion by Quincy Robinson and Paul Martyn-Smith (Evidence of Modern Physical Knowledge from Asiatic Antiquity: Re-integration: Nine Realms of Middle Earth, 2015).

Traditional derivation of magic square organization from the Lo Shu Turtle
Example of Lo Shu Turtle magic square configuration	Example of the simplest magic square (of order 3)	Correspondences between Lo Shu, Ba Gua and 3x3 magic square patterns

The 3x3 table of numbers (above centre) could be usefully recognized as an array of boxes (notably characteristic of modern office architecture and knowledge management), namely the constraining contexts associated with the perception of the frog in the Zhuangzi fable -- as discussed above with respect to in-the-box cognition. Transcendence of those constraints is however only then achieved by the "magic" of the arrangement, as the "pattern which connects". It is in this sense that the turtle (or eagle) can be unconstrained, being able to move freely in relation to the limitations of the box configurations -- effectively, as by magic, through its walls. This freedom is evident with some pieces on a chess board -- in contrast to the constraints on others.

The significance of any such pattern follows from the much-cited phrase of Gregory Bateson (Mind and Nature: a necessary unity, 1979), variously discussed separately:

• Candice Bradley: Gregory Bateson: The Pattern Which Connects,
• Paul Clarke: The Pattern Which Connects (School of Sustainability, 1 February 2017)
• Richard J. Borden: Gregory Bateson's Search for "Patterns Which Connect" Ecology and Mind (Human Ecology Review,  23, 2017, 2)
• Roy C. Dudgeon: The Pattern Which Connects: Batesonian Holism and Postmodern Science
• Søren Brier: Bateson and Peirce on the Pattern that Connects and the Sacred (A Legacy for Living Systems, 2008)
• Paul Andersen and David Salomon: The Pattern that Connects (Australian Humanities Review, 35, 2005, June)
• Overriding Law: a pattern that connects (Insights, 17 September 2020).

Identification of such pattern is now a feature of explorations of pattern language  Allusions to the cognition of the turtle of the fable might then be associated with references to the hyperspace, hyperreality and hyperobjects of modern insights -- as traditionally framed (Hyperspace Clues to the Psychology of the Pattern that Connects, 2003). The latter is explored in the light of the 81 Tao Te Ching insights.

Any such cognitive possibility merits consideration of the potential role of such a configuration, as discussed separately (Salvation Enabled by Systemic Comprehension -- via aesthetics of magic squares? 2015). The possibility was explored in terms of the following:

Although often held to be a mathematical curiosity, of relevance to that discussion is the importance it was held to have been a preoccupation of Benjamin Franklin, one of the Founding Fathers of the United States, a drafter and signer of the United States Declaration of Independence. How it may have influenced the design and memorability of the latter remains unclear (Memorability, Mnemonics, Maths, Music and Governance , 2002).

Benjamin Franklin's 8x8 magic squares animations of movement of selected bent diagonals
Vertical movement	Combined movement	Horizontal movement

Greater "magic"? Franklin called his 16x16 magic square the most magically magical of any magic square ever made by a magician -- with which many mathematicians and mystics would now be held to agree (Peter Loly, Franklin Squares: a chapter in the scientific studies of magical squares, University of Manitoba, 2006; William H. Richardson, Ben Franklin's Amazing Magic Square [including animation], Wichita State University; Ben Franklin's 8x8 Magic Square, Wichita State University).

The methods by which he generated such squares so readily, characterized by so-called bent diagonals, remain unknown (Harvey Heinz, Most-perfect Bent diagonal Magic Squares, 2009; Daniel Schindel, et al., Enumerating the bent diagonal squares of Dr Benjamin Franklin, Proceedings of the Royal Society, 462, 2006), pp. 2271-2279; Paul Pasles (The Lost Squares of Dr. Franklin, The American Mathematical Monthly, 108, 2001, 6).

Benjamin Franklin's 16x16 magic squares animations of movement of selected bent diagonals
Vertical movement	Combined movement	Horizontal movement

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]